Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre

Let a, b and c be positive integers and consider a hexagon with side lengths a,b,c,a,b,c whose angles are 120◦ (see Figure 1). The subject of our interest is rhombus tilings of such a hexagon using rhombi with all sides of length 1 and angles 60◦ and 120◦. Figure 2 shows an example of a rhombus tiling of a hexagon with a = 3, b = 5 and c = 4. A first natural question to be asked is how many rhombus tiling of a fixed hexagon exist. A well known bijection between such rhombus tilings and plane partitions contained in an a× b× c box [3] and MacMahon’s enumeration of plane partitions [11, Sec. 429, q → 1; proof in Sec. 494] give the following answer: The number of all rhombus tilings of a hexagon with side lengths a,b,c,a,b,c equals